Early on, when solving the TDDFT equation was computationally intractable,
there were many attempts to extract from the TDDFT equation a collective model
that could be adapted to describing large-amplitude collective motion. The
adiabatic time-dependent Hartree-Fock (ATDHF) and its extension to include
pairing correlations (ATDHFB) provided many insights into such an approach
[1]}, [2]}, [3]}, [4]}. Here, we only outline
the main concepts of these theories and focus on ATDHFB; additional details can
be found in the aforementioned references as well as the more recent reviews
[5]}, [6]}. In TDHFB, the system explores a large
variational space during its evolution; in ATDHFB, one seeks to force the
system to follow specific trajectories within a collective subspace \(\Sigma \)  of
the entire Hilbert space which, ideally, is decoupled from the rest of the
space. In the seminal paper of Baranger and Veneroni on ATDHF
[4]}, this was achieved by introducing a unitary transform of
the TDHF density \(\rho (t)\) . In the case of ATDHFB, this transformation reads:
\(\mathcal {R}(t)\rightarrow e^{i\hat{\chi }(t)}\mathcal {R}(t)e^{-i\hat{\chi }(t)}\) 
where \(\hat{\chi }(t)\)  is a one-body operator that is supposed to be small. The
transformed density is expanded up to second order in \(\hat{\chi }\)  together
with the time-dependent HFB matrix, and the results are inserted into the TDHFB
equation. Separating contributions that are time-odd from those that are
time-even, we obtain the two coupled equations

iR(0)  = 
[ H(0), R(1) ]
+
[ H(1), R(0) ] ,
eq:atdhfbeq1
